Radial waves in a borehole and stoneley waves for measuring formation permeability and electroacoustic constant

ABSTRACT

A Stoneley wave is generated in a borehole in a saturated porous earth formation. Measurements are made of the velocity of motion of the formation and the fluid in the formation. The difference in the velocities is indicative of formation permeability. An additional measurement of the electric field at the borehole wall enables determination of the electroacoustic constant.

BACKGROUND OF THE DISCLOSURE

1. Field of the Disclosure

The present disclosure pertains to logging while drilling apparatus andmore particularly to acoustic logging while drilling apparatus andgenerating and using Stoneley waves to measure formation permeabilityand electroacoustic constant.

2. Summary of the Related Art

Acoustic wellbore logging instruments are used to measure velocities ofearth formations in one or more modes of acoustic energy propagation.Acoustic wellbore logging instruments are typically used inliquid-filled wellbores drilled through the earth formations. Velocityis generally determined using these instrument by measuring the timetaken by an acoustic energy pulse to traverse a particular distancealong the wall of the wellbore. The wall forms the interface between theliquid in the wellbore and the earth formations.

One form of acoustic energy pulses of particular interest to theinvention is referred to as “Stoneley” waves. Stoneley waves, also knownas tube waves, are essentially guided pressure pulses travelling in thewellbore. It had been determined in earlier research that a relationshipis likely to exist between the transmission properties of Stoneley wavesas they propagate along the wellbore, and the hydraulic characteristicsof the formations along the wellbore wall. See for example, J.Rosenbaum, Synthetic Microseismograms: Logging in Porous Formations,Geophysics, vol. 39, pp. 14-32, Society of Exploration Geophysicists(1974). Determining formation permeability was not practical using theacoustic logging instruments available at the time the Rosenbaumreference was published because those instruments typically did notgenerate detectable Stoneley waves, and in the instances where they didso, separation of the Stoneley waves from the acoustic signals asdetected was very difficult.

U.S. Pat. No. 5,784,333 to Tang et al., having the same assignee as thepresent disclosure and the contents of which are incorporated herein byreference, discloses a method for determining the permeability of earthformations penetrated by a wellbore from acoustic signals measured by anacoustic wellbore logging instrument. The method includes separatingcomponents from the measured acoustic signals which represent Stoneleywaves propagating through the earth formations. Signals representingStoneley waves propagating through the same earth formations aresynthesized. The separated acoustic signal components and thesynthesized Stoneley wave signals are compared. The permeability isdetermined from differences between the synthesized Stoneley wavesignals and the separated acoustic signal components. In a preferredembodiment, the step of calculating the permeability includes inversionprocessing a wave center frequency shift and a wave travel time delaywith respect to the permeability of the earth formations.

The present disclosure is directed towards a method and apparatus fordirectly measuring the formation permeability and the electroacousticconstant of a porous formation. It makes use of the fact that an elasticwave (such as a Stoneley wave) propagating along the borehole wallproduces an electrical signal that can be measured.

SUMMARY OF THE DISCLOSURE

One embodiment of the disclosure is a method of evaluating an earthformation. The method includes: producing a state associated with avelocity of motion of a fluid in the borehole and a velocity of motionof the formation at an interface between the fluid and the formationsuch that a difference between the velocity of motion of the fluid inthe borehole and the velocity of motion of the formation is dependentupon permeability; and using a processor to estimate an electroacousticconstant of the earth formation using a first measurement indicative ofan electric charge on a wall of the borehole, wherein the firstmeasurement is taken during the state. Producing the state may includeusing a swept frequency source in a borehole and identifying at leastone frequency of the source producing the state by identifying from aplurality of frequencies in a frequency sweep a frequency of the sourceat which a corresponding difference between a corresponding fluidvelocity in the borehole and a corresponding velocity of the formationis at a maximum.

Another embodiment of the disclosure is a system for evaluating an earthformation. The system includes: a swept frequency source on a toolconfigured to be conveyed into a borehole and generate a wave in a fluidin the borehole over a plurality of frequencies; a first sensorconfigured to make a first measurement indicative of a charge on a wallof the borehole at the plurality of frequencies; a second sensorconfigured to make a second measurement indicative of a velocity of thefluid in the borehole at the plurality of frequencies; a third sensorconfigured to make a third measurement indicative of a velocity of thewall of the borehole at the at least one resonant frequency; and aprocessor configured to: identify at least one resonant frequency of theborehole fluid within the plurality of frequencies, and use the first,second and third measurements at the at least one resonance frequency toestimate an electroacoustic constant of the earth formation. Theprocessor may be further configured to estimate a permeability of theearth formation using an additional first measurement and additionalmeasurements of the fluid velocity and the velocity of the formation atat least one additional frequency different from the at least oneresonant frequency.

BRIEF DESCRIPTION OF THE FIGURES

For detailed understanding of the present disclosure, references shouldbe made to the following detailed description of exemplaryembodiment(s), taken in conjunction with the accompanying drawings, inwhich like elements have been given like numerals, wherein:

FIG. 1 shows an acoustic well logging instrument as it is used toacquire signals useful with the method of this invention;

FIG. 2 is an illustration of an exemplary configuration of acoustictransmitters, a flow sensor, a geophone and an electrical charge sensor;

FIG. 3 is a plot of the difference between the velocities of thesaturating fluid and the porous matrix a function of frequency;

FIG. 4 shows an exemplary dependence of the difference in velocities ofthe saturating fluid and the porous matrix as a function ofpermeability; and

FIG. 5 shows the dependence of the ratio of the electric field to thevelocity of deformation of the borehole upon permeability for differentfrequencies.

DETAILED DESCRIPTION OF THE EMBODIMENTS

FIG. 1 shows an acoustic well logging instrument as it is used toacquire signals suitable for processing according to the method of thisdisclosure. The instrument 10 is inserted into a wellbore 2 drilledthrough earth formations 20. The instrument 10 can be inserted into andwithdrawn from the wellbore 2 by means of an armored electrical cable 14spooled from a winch (not shown) or any similar conveyance known in theart.

The wellbore 2 is typically filled with a liquid 4, which can be“drilling mud” or any similar fluid usually used for drilling orcompletion of wellbores. The instrument includes a plurality of acoustictransmitter 8. The transmitters 8 are periodically energized and emitacoustic energy that radiate from the tool 10. The instrument 10typically includes a telemetry module, shown generally at 12, whichconverts the electrical signals from the sensors (discussed withreference to FIG. 2) into a form suitable for recording and transmissionover the cable 14 to a surface processor 18. It should be understoodthat the number of transmitters 8 as shown in FIG. 1 is only an exampleand is not meant to limit the invention.

Shown in FIG. 2 are an array of transmitters 241 a . . . 241 n. Thearrays may include elements that are axially and/or circumferentiallydisposed. While the illustration shows them on a single housing, this isnot to be construed as a limitation to the disclosure; a commonconfiguration uses transmitters and receivers on more than one sub. Alsoshown in the figure is the borehole 26, and the logging tool 100. Alsoshown are a sensor 271 to measure the electric charge or field at theborehole wall, a sensor 273, such as a flow rate sensor, to measure theborehole fluid velocity and a motion sensor such as a geophone 275 tomeasure the velocity of the deformation of the borehole wall. Thepresent disclosure includes two methods of measuring the surface charge(or the electric field in the porous medium). One is to connect anammeter to the borehole wall as sensor 271 with the other end of theammeter grounded: the measured current is indicative of the surfacecharge. A second method is drill pilot holes 226 in the formation andmeasure the potential difference between the borehole wall and thebottom of the pilot hole using a sensor 277 in the bottom of the pilothole 226.

It should be noted that the disclosure is with reference to a wirelineconveyed logging tool. This is not to be construed as a limitation, andthe transmitter array for generating Stoneley waves could be on abottomhole assembly (BHA) and used for imaging ahead of the borehole.This is discussed in U.S. Pat. No. 8,055,448 of Mathiszik et al., havingthe same assignee as the present disclosure and the contents of whichare incorporated herein by reference.

With a monopole excitation of the elements of the transmitter array 241a . . . 241 n, radial excitation of the fluid in the borehole results.The theory of the radial oscillations, as it is shown below, shows thatthe electric charge appearing on the surface of the interface isdetermined by the following ratio:

$\begin{matrix}{\left. Q \right|_{boundary} = {\frac{\left. E_{r} \right|_{boundary}}{4\pi} = {\alpha \cdot {\frac{\left. {\rho_{l}\left( {v_{r} - u_{r}} \right)} \right|_{boundary}}{4\pi\;\sigma}.}}}} & (1)\end{matrix}$In the above formula, α is the electroacoustic constant of the porousmedium (the earth formation), σ is electric conductivity, ρ_(l) ispartial density of fluid in the saturated porous medium (earthformation), v_(r)−u_(r) is the difference between velocities of thesaturating fluid and the porous matrix in the radial oscillations modeoutside of the resonance zones. Having measured the difference betweenvelocities v_(r)−u_(r) (to this end, we could perform independentmeasurements of the fluid velocity v_(r) ⁽⁰⁾ and velocity u_(r) ⁽¹⁾ ofporous matrix next to the interface between the media, and based onthese data, calculate the difference between velocities u_(r) ⁽¹⁾−v_(r)⁽¹⁾=(u_(r) ⁽¹⁾−v_(r) ⁽⁰⁾)/φ) and the surface electric charge using aspecial sensor, we can calculate the electroacoustic constant via theformula (1) above:

$\begin{matrix}{\alpha = {\frac{\left. {4{\pi\sigma}\; Q} \right|_{boundary}}{\left. {\rho_{l}\left( {v_{r} - u_{r}} \right)} \right|_{boundary}}.}} & (2)\end{matrix}$In order to estimate the electroacoustic constant using this method, themethod includes a measurement of formation porosity φ using, forexample, a neutron porosity logging tool. The partial density of theliquid ρ_(l) is given by the product φρ_(l) ^(ph) where the physicalfluid density ρ_(l) ^(ph) is measured by a density logging tool, such asa gamma ray density tool. The conductivity of the formation fluid σ maybe measured by known methods.

An important comment should be made here. In the first approximation,the difference v_(r)−u_(r) does not depend on permeability of the porousspace k. FIG. 3 shows the dependence of the difference betweenvelocities v_(r)−u_(r) 301 (ordinate) upon the frequency of the signal(abscissa) for an exemplary model. The model parameters are: boreholeliquid density ρ_(f)=1.0 g/cm³; sound velocity in borehole liquidc_(p0)=1.45·10⁵ cm/s; porous matrix density ρ_(s)=2.2 g/cm³; saturatedliquid density ρ_(l)=1.0 g/cm³; porosity φ=0.2; first longitudinal soundvelocity c_(p1)=2.2·10⁵ cm/s; second longitudinal sound velocityc_(p2)=1.2·10⁵ cm/s; share sound velocity c_(t)=1.6·10⁵ cm/s; liquidviscosity μ=1.05·10⁻² P; permeability k=50 mD; source radius r₁=1.0 cm;borehole radius r₂=10 cm.) The intersection of 301 with the line 303gives an indication of the width of the resonance zones. The derivationof the curves follows from the theory presented below, discussed inAppendix I.

For the case where the formation is impermeable, then the configurationshown in FIG. 2 may be recognized as a waveguide for which the resonancefrequencies depend upon the radius of the tool, the radius of theborehole and the velocity of compressional waves in the borehole. Whenthe formation has a non-zero permeability, the configuration in FIG. 2acts as a leaky waveguide with the leakage characteristics beingdependent upon the permeability. This is the basis for the determinationof formation permeability in the present disclosure.

FIG. 4 shows the dependence of the resonance on permeability. Theabscissa is the permeability. The curve 401 shows the dependence of thedifference in velocities on the permeability in a narrow resonance zoneabout 300 Hz wide at a frequency of 27.39 kHz. The curve 403 shows thedependence of the difference in velocities on permeability outside theresonance zone. Thus, outside of the resonance zone, the differencebetween velocities depends weakly on permeability, and we may assume, inthe first approximation, that the surface electric charge formed on theinterface does not depend on permeability as the surface electric chargeis a function primarily of the electroacoustic constant. Similar resultscan be obtained for the other resonances in FIG. 3. We next discuss theelectroacoustic ratio of the Stoneley wave.

As derived in the theory presented below in Appendix II, for theStoneley wave, the ratio of the electric field E_(z) amplitude to theamplitude V_(z) in the acoustic wave which determined this field has thefollowing form:

$\begin{matrix}{{\frac{E_{z}}{V_{z}} = {\alpha \cdot k \cdot {f\left( {\alpha,k} \right)}}},} & (3)\end{matrix}$where ƒ(α, k) is a function weakly dependent on its arguments. Thisdependence is illustrated in FIG. 5. Shown therein is a plot of

$\frac{E_{z}}{V_{z}}$as a function of permeability for frequencies of 30 kHz 501, 15 kHz 503and 3 kHz 505.

When permeability changes by three orders of magnitude, the functionchanges only by the factor of two. Having the theoretical value of f(α,k), measured value of E_(z)/V_(z), and computed value of α (see eqn.(2)), we can find permeability:

$\begin{matrix}{k = {\frac{E_{z}/V_{z}}{\alpha\;{f\left( {\alpha,k} \right)}}.}} & (4)\end{matrix}$Thus, having performed two types of measurements: one with the radialwaves and one with the electroacoustic ratio of the Stoneley wave, wefirst measure the electroacoustic constant and then, permeability. Inorder to improve accuracy of subsequent measurements, in one embodimentof the disclosure, an iterative procedure is used taking into accountthe weak functional dependence ƒ(α,k) and (v_(r)−u_(r))|_(boundary)(α,k). In such an iterative process, a first estimate of permeability ismade ignoring the functional dependence, and the first estimate is usedto calculate the functional dependence of αf(α, k) and a revised valueof the permeability is calculated.

The description above has been in terms of a device conveyed on a BHA ona drilling tubular into a borehole in the earth formation. The methodand apparatus described above could also be used in conjunction with alogging string conveyed on a wireline into the earth formation. For thepurposes of the present disclosure, the BHA and the logging string maybe referred to as a “downhole assembly.” It should further be noted thatwhile the example shown depicted the transmitter assembly and thereceiver assembly on a single tubular, this is not to be construed as alimitation of the disclosure. It is also possible to have a segmentedacoustic logging tool to facilitate conveyance in the borehole. Once theformation permeability has been estimated, it can be used for furtherreservoir development operations.

Implicit in the processing of the data is the use of a computer programimplemented on a suitable machine readable medium that enables theprocessor to perform the control and processing. The machine readablemedium may include ROMs, EPROMs, EAROMs, Flash Memories and Opticaldisks. The determined formation velocities and boundary locations may berecorded on a suitable medium and used for subsequent processing uponretrieval of the BHA. The determined formation permeability andelectroacoustic constant may further be telemetered uphole for displayand analysis.

APPENDIX I

To measure permeability in the saturated porous medium, a theory thatdescribes the radial waves based on the linearized version of thefiltration theory is used. To describe the acoustic field in fluid, thefollowing equation is used:{umlaut over (v)}−c _(p0) ²∇ div v=0.  (I.1)Here c_(p0) is the velocity of sound in borehole fluid. The electricalfield in porous medium is defined by

$\begin{matrix}{{E = {{- \frac{\alpha\;\rho_{l}}{\sigma}}\left( {u - v} \right)}},} & \left( {I{.2}} \right)\end{matrix}$where α is the electroacoustic constant of the porous medium, and σ isthe electric conductivity of the porous medium. Velocities u, v of thematrix of the porous medium and the fluid contained therein satisfyequations

$\begin{matrix}{{{\overset{¨}{u} - {c_{t}^{2}\Delta\; u} - {a_{1}{\nabla{div}}\; u} + {a_{2}{\nabla{div}}\; v} + {\frac{\rho_{l}}{\rho_{s}}{b\left( {\overset{.}{u} - \overset{.}{v}} \right)}}} = 0},} & \left( {I{.3}} \right) \\{{\overset{¨}{v} + {a_{3}{\nabla{div}}\; u} - {a_{4}{\nabla{div}}\; v} - {b\left( {\overset{.}{u} - \overset{.}{v}} \right)}} = 0.} & \left( {I{.4}} \right)\end{matrix}$Here ρ_(l), ρ_(s) are partial densities of the saturating fluid and theporous matrix, respectively, ρ=ρ_(l)+ρ_(s) is density of the saturatedporous medium. The dissipative coefficient is b=ρ_(l)χ=η/(kρ), where ηis dynamic viscosity the saturating fluid, k is permeability.

Equation coefficients α_(i) (j=1, . . . , 4) are determined by theelasticity moduli λ, μ, and γ:

$\begin{matrix}{{a_{1} = {\frac{1}{\rho_{s}}\left( {{\frac{\rho_{s}^{2}}{\rho^{2}}\gamma} + {\frac{\rho_{l}^{2}}{\rho^{2}}K} + {\frac{1}{3}\mu}} \right)}},{a_{2} = {\frac{\rho_{l}}{\rho_{s}}\left( {{\frac{\rho_{l}}{\rho^{2}}K} - {\frac{\rho_{s}}{\rho^{2}}\gamma}} \right)}},} & \left( {I{.5}} \right) \\{{a_{3} = {{\frac{\rho_{l}}{\rho^{2}}K} - {\frac{\rho_{s}}{\rho^{2}}\gamma}}},{a_{4} = {{\frac{\rho_{l}}{\rho^{2}}K} + {\frac{\rho_{l}}{\rho^{2}}\gamma}}},} & \left( {I{.6}} \right)\end{matrix}$where K=λ+2μ/3. Three elasticity moduli μ, λ, and γ are determinedunivalently by three measurable velocities of sound in saturated porousmedium (two compressional velocities c_(p1), c_(p2) and one shearvelocity c_(t))

$\begin{matrix}{\mspace{79mu}{{\mu = {\rho_{s}c_{t}^{2}}},}} & \left( {I{.7}} \right) \\{{K = {\frac{1}{2}\frac{\rho_{s}}{\rho_{l}}\left( {{\rho\; c_{p\; 1}^{2}} + {\rho\; c_{p\; 2}^{2}} - {\frac{8}{3}\rho_{l}c_{t}^{2}} - \sqrt{\left( {{\rho\; c_{p\; 1}^{2}} - {\rho\; c_{p\; 2}^{2}}} \right)^{2} - {\frac{64}{9}\rho_{s}\rho_{l}c_{t}^{4}}}} \right)}},} & \left( {I{.8}} \right) \\{\gamma = {\frac{1}{2}{\left( {{\rho\; c_{p\; 1}^{2}} + {\rho\; c_{p\; 2}^{2}} - {\frac{8}{3}\rho_{s}c_{t}^{2}} + \sqrt{\left( {{\rho\; c_{p\; 1}^{2}} - {\rho\; c_{p\; 2}^{2}}} \right)^{2} - {\frac{64}{9}\rho_{s}\rho_{l}c_{t}^{4}}}} \right).}}} & \left( {I{.9}} \right)\end{matrix}$

The acoustic source in the system is a finite-size harmonic pressuregenerator p=P₀e^(−iωt) located in the center of the wellbore. Theacoustic response of the system under study is also harmonic (u, v)→(u,v)e^(−iωt). The latter fact enables us to keep the following coordinatedependence of equations (I.1, I.2, I.3) in Fourier amplitudes of theradial wave velocities u(u_(r), 0, 0), v(v_(r), 0, 0):λ₀∇ div v+v=0  (I.10)Andā ₀ Δu−ā ₂ Δv+(1+εb )u−εbv=0, (r ₂ <r<∞),  (I.11)−ā ₃ Δu+ā ₄ Δv−bu+(1+b )v=0, (r ₂ <r<∞).  (I.12)Here we have dimensionless equations where velocities u, v are in unitsof c_(t), coordinates are in units of c_(t)/ω; coefficients aretransformed: λ₀=c_(p0) ²/c_(t) ², ā_(i)=a_(i)/c_(t) ² (i=0, . . . , 4),b=ib/ω; ε=ρ_(l)/ρ_(s), ā₀=ā₁+1.

The electrical field in dimensionless form is defined by

$\begin{matrix}{{E = {{- \frac{c_{e}}{c_{t}}}\left( {u - v} \right)}},} & \left( {I{.13}} \right)\end{matrix}$where c_(e) is the velocity of light, E is in units of αc_(e)ρ_(l)/σ.Let us find common solutions to equations (I.10-I.12).

To reduce the equations described the acoustic field in porous medium tothe diagonal form, the set (I.11-I.12) could be conveniently expressedin the matrix form.AΔU+WU=0  (I.14)Here

$\begin{matrix}{{U = \begin{pmatrix}u_{r} \\v_{r}\end{pmatrix}},{A = \begin{pmatrix}{\overset{\_}{a}}_{0} & {- {\overset{\_}{a}}_{2}} \\{- {\overset{\_}{a}}_{3}} & {\overset{\_}{a}}_{4}\end{pmatrix}},{W = {\begin{pmatrix}\left( {1 + {ɛ\;\overset{\_}{b}}} \right) & {{- ɛ}\;\overset{\_}{b}} \\{- \overset{\_}{b}} & \left( {1 + \overset{\_}{b}} \right)\end{pmatrix}.}}} & \left( {I{.15}} \right)\end{matrix}$Having multiplied set (I.12) by the inverse matrix

${W^{- 1} = \begin{pmatrix}\frac{\left( {1 + \overset{\_}{b}} \right)}{\det\; W} & \frac{ɛ\overset{\_}{b}}{\det\; W} \\\frac{\overset{\_}{b}}{\det\; W} & \frac{\left( {1 + {ɛ\overset{\_}{b}}} \right)}{\det\; W}\end{pmatrix}},$where det W=1+(1+ε)b, we arrive at the set of equations:HΔU+U=0  (I.16)with this matrix of coefficients: H=W⁻¹Ah ₁₁=((1+ b )ā ₀ −εbā ₃)/det W, h ₁₂=(−(1+ b )ā ₂ +εbā ₄)/det W,h ₂₁=( bā ₀−(1+ε b )ā ₃)/det W, h ₂₂=(− bā ₂+(1+ε b )ā ₄)/det W.  (I.17)

To reduce set (I.14) to the diagonal form, we multiply it by the matrix

${R = \begin{pmatrix}R_{1} & R_{2} \\R_{3} & R_{4}\end{pmatrix}},$as found from this problem solution for the eigenvalues of the matrixH:HR=RΛ  (I.18)as the matrix compiled of the right-hand side eigenvectors,

$R:{\left( {\begin{pmatrix}R_{1} \\R_{3}\end{pmatrix}\begin{pmatrix}R_{2} \\R_{4}\end{pmatrix}} \right).}$Components of the transform matrix R (or eigenvectors of the matrix H)have the following:R ₁=1, R ₂=(λ₂ −h ₂₂)/h ₂₁ , R ₃ =h ₂₁/(λ₁ −h ₂₂), R ₄=1.  (I.19)

The matrix of eigenvectors

$\Lambda = \begin{pmatrix}\lambda_{1} & 0 \\0 & \lambda_{2}\end{pmatrix}$is diagonal, so that R⁻¹H R=Λ, and eigenvalues λ₁, λ₂ can be expressedas follows:λ₁=½tr H+√{square root over (½(tr H)²−det H)}, λ₂=½tr H−√{square rootover (½(tr H)²−det H)},  (I.20)where tr H=h₁₁+h₂₂ is the trace of the matrix H, det H=h₁₁h₂₂−h₁₂h₂₁ isthe determinant of matrix H.The vector-column of functions is transformed as follows:

$\begin{matrix}{{{R^{- 1}u} = V},{{{where}\mspace{14mu} V} = {\begin{pmatrix}v_{1} \\v_{2}\end{pmatrix}.}}} & \left( {I{.21}} \right)\end{matrix}$After transformation, set (I.16) becomes diagonal:ΛΔV+V=0,  (I.22)or, in coordinate notations (prime denotes the derivative with respectto the radius):

$\begin{matrix}{{v_{1}^{''} + {\frac{1}{r}v_{1}^{\prime}} + {\left( {\frac{1}{\lambda_{1}} - \frac{1}{r^{2}}} \right)v_{1}}} = 0} & \left( {I{.23}} \right) \\{{v_{2}^{''} + {\frac{1}{r}v_{2}^{\prime}} + {\left( {\frac{1}{\lambda_{2}} - \frac{1}{r^{2}}} \right)v_{2}}} = 0} & \left( {I{.24}} \right)\end{matrix}$

The initial velocities can be expressed via functions v₁, v₂ (I.21):u _(r) =v ₁ +R ₂ v ₂ , v _(r) =R ₃ v ₁ +v ₂.  (I.25)

The Acoustic equation (I.10) for the borehole fluid can be expressedsimilarly:

$\begin{matrix}{{v_{r}^{''} + {\frac{1}{r}v_{r}^{\prime}} + {\left( {\frac{1}{\lambda_{0}} - \frac{1}{r^{2}}} \right)v_{r}}} = 0.} & \left( {I{.26}} \right)\end{matrix}$The set of equations (I.23, I.24, I.26) taking into account (I.25)describes the acoustic field of the radial waves in the wellbore and inthe formation beyond the wellbore. The boundary conditions for thisgeometry (in the dimensionless case) are the following (the index inbrackets denotes values pertaining to the fluid ⁽⁰⁾ and the saturatedporous medium ⁽¹⁾):

-   1) On the surface of the source exciting the oscillations in the    fluid, (r=r₁, FIG. 2), a harmonic source is set:    p ⁽⁰⁾ =P ₀  (I.27)-   2) On the interface between the borehole and the saturated porous    medium r=r₂, FIG. 2), continuity of the medium is assumed, as well    as continuity of the full stress tensors (Σ_(rr) ⁽¹⁾, Σ_(rr) ⁽⁰⁾)    and the partial pressures:    (1−φ)u _(r) ⁽¹⁾ +φv _(r) ⁽¹⁾ =v _(r) ⁽⁰⁾,  (I.28)

$\begin{matrix}{{\sum\limits_{rr}^{(1)}{= \overset{(0)}{\sum\limits_{rr}}}},} & \left( {I{.29}} \right) \\{{\frac{p^{(1)}}{\rho^{(1)}} - {\kappa\;\frac{\rho^{(1)}}{\rho_{s}^{(1)}}{\overset{\_}{b}\left( {\varphi_{u}^{(1)} - \varphi_{v}^{(1)}} \right)}}} = {\frac{p^{(0)}}{\rho^{(0)}}.}} & \left( {I{.30}} \right)\end{matrix}$Here φ is porosity, φ_(u,v) are potentials which determine the velocityof the radial waves in the medium u=∇φ_(u), v=∇φ_(v), κ is a conditionalcoefficient which enables us to change the friction force at theboundary, P ₀=P₀/(ρ_(s)c_(t) ²).

The final boundary condition follows from a more general continuitycondition for the partial stress tensor in fluid.Σ_(rr) ^(l(1))=φΣ_(rr) ⁽⁰⁾,  (I.30′)where Σ_(rr) ⁽¹⁾=Σ_(rr) ^(s(1))+Σ_(rr) ^(l(1)).Components of stress tensors and pressure are determined viadisplacement velocities as follows:−{dot over (p)} ⁽¹⁾=π₁ div u ⁽¹⁾+π₂ div v ⁽¹⁾,  (I.31){dot over (Σ)}_(rr) ⁽¹⁾=π₃ div u ⁽¹⁾+π₄ div v ⁽¹⁾+2u _(r)′,  (I.32){dot over (Σ)}_(rr) ⁽⁰⁾ =−{dot over (p)} ⁽⁰⁾=π₀ div v ⁽⁰⁾,  (I.33)where the following is determined:

$\begin{matrix}{{\pi_{1} = {\left( {{\frac{\rho_{s}}{\rho}\gamma} - {\frac{\rho_{l}}{\rho}K}} \right)/\left( {\rho_{s}c_{t}^{2}} \right)}},{\pi_{2} = {\left( {{\frac{\rho_{s}}{\rho}\gamma} - {\frac{\rho_{l}}{\rho}K}} \right)/\left( {\rho_{s}c_{t}^{2}} \right)}},} & \left( {I{.34}} \right) \\{{\pi_{3} = {\left( {{\frac{\rho_{s}}{\rho}\gamma} - {\frac{2}{3}\mu}} \right)/\left( {\rho_{s}c_{t}^{2}} \right)}},{\pi_{4} = {\left( {\frac{\rho_{l}}{\rho}\gamma} \right)/\left( {\rho_{s}c_{t}^{2}} \right)}},} & \left( {I{.35}} \right) \\{\pi_{0} = {\rho^{(0)}{c_{p\; 0}^{2}/{\left( {\rho_{s}c_{t}^{2}} \right).}}}} & \left( {I{.36}} \right)\end{matrix}$

The set of equations (I.23, I.24, I.26) with boundary conditions(I.27-I.30) for geometry shown in FIG. 1 has a solution that isexpressed via the Hankel functions H_(α) ⁽¹⁾(z), H_(α) ⁽²⁾(z). In thesaturated porous medium, this solution in a dimensionless form is

for pressure:p ⁽⁰⁾ −iB ₁(π₁+π₂ R ³)H ₁ ⁽¹⁾(r/√{square root over (λ₁)})/√{square rootover (λ₁)}−iB ₂(π₁ R ₂+π₂)H ₁ ⁽¹⁾(r/√{square root over (λ₂)})/√{squareroot over (λ₂)},  (I.37) andfor the solid matrix velocities and velocities of the saturating fluid:u _(r) ⁽¹⁾ =B ₁ H ₁ ⁽¹⁾(r/√{square root over (λ₁)})+B ₂ R ₂ H ₁⁽¹⁾(r/√{square root over (λ₂)}),  (I.38)v _(r) ⁽¹⁾ =B ₁ R ₃ H ₁ ⁽¹⁾(r/√{square root over (λ₁)})+B ₂ H ₁⁽¹⁾(r/√{square root over (λ₂)}),  (I.39)and, consequently, for the difference of the matrix and fluidvelocities:u _(r) ⁽¹⁾ −v _(r) ⁽¹⁾ =B ₁(1−R ₃)H ₁ ⁽¹⁾(r/√{square root over (λ₁)})+B₂(R ₂−1)H ₁ ⁽¹⁾(r/√{square root over (λ₂)}),  (I.40)and electrical field

$\begin{matrix}{E_{r}^{(1)} = {{- \frac{c_{e}}{c_{t}}}{\left( {u_{r}^{(1)} - v_{r}^{(1)}} \right).}}} & \left( {I{.41}} \right)\end{matrix}$

In the wellbore, this solution in a dimensionless form is:

for pressure in fluidp ⁽⁰⁾ =−iB ₀π₀ H ₁ ⁽¹⁾(r/√{square root over (λ₀)})/√{square root over(λ₀)}−iC ₀π₀ H ₁ ⁽²⁾(r/√{square root over (λ₀)})/√{square root over(λ₀)},  (I.42)for the fluid velocityv _(r) ⁽⁰⁾ =B ₀ H ₁ ⁽¹⁾(r/√{square root over (λ₀)})+C ₀ H ₁⁽²⁾(r/√{square root over (λ₀)}).  (I.43)Integration constants B₁, B₂, B₀, C₀ are determined by the conditionthat boundary problem solutions (I.20, A.21, A.23) are to satisfyboundary conditions (I.24-I.27). Substituting solutions (I.34-I.40) intoboundary conditions (I.24-I.27), we arrive at this set:GΞ=P,  (I.44)where one can see the column vector of the integration constants Ξ, thesource part P, and the coefficient matrix G:

$\begin{matrix}{{\Xi = \begin{pmatrix}B_{1} \\B_{2} \\B_{0} \\C_{0}\end{pmatrix}},{P = \begin{pmatrix}{\overset{\_}{P}}_{0} \\0 \\0 \\0\end{pmatrix}},{G = \begin{pmatrix}g_{11} & g_{12} & g_{13} & g_{14} \\g_{21} & g_{22} & g_{23} & g_{24} \\g_{31} & g_{32} & g_{33} & g_{34} \\g_{41} & g_{42} & g_{43} & g_{44}\end{pmatrix}},} & \left( {I{.45}} \right)\end{matrix}$Whereg ₁₁=0, g ₁₂=0,  (I.46)g ₁₃=π₀ H ₀ ⁽¹⁾(r ₁/√{square root over (λ₀)})/√{square root over (λ₀)},g ₁₄=π₀ H ₀ ⁽²⁾(r ₁/√{square root over (λ₀)})/√{square root over(λ₀)},  (I.47)g ₂₁=((1−φ)+φR ₃)H ₁ ⁽¹⁾(r ₂/√{square root over (λ₁)}), g ₂₂=((1−φ)R₂+φ)H ₁ ⁽¹⁾(r ₂/√{square root over (λ₂)}),  (I.48)g ₂₃ =−H ₁ ⁽¹⁾(r ₂/√{square root over (λ₀)}), g ₂₄ =−H ₁ ⁽²⁾(r₂/√{square root over (λ₀)}),  (I.49)g ₃₁=(π₃+π₄ R ₃+2)H ₀ ⁽¹⁾(r ₂/√{square root over (λ₁)})/√{square rootover (λ₁)}−2H ₁ ⁽¹⁾(r ₂/√{square root over (λ₁)})/r ₂,  (I.50)g ₃₂=(π₃ R ₂+π₄+2)H ₀ ⁽¹⁾(r ₂/√{square root over (λ₂)})/√{square rootover (λ₂)}−2R ₂ H ₁ ⁽¹⁾(r ₂/√{square root over (λ₂)})/r ₂,  (I.51)g ₃₃=−π₀ H ₀ ⁽¹⁾(r ₂/√{square root over (λ₀)})/√{square root over (λ₀)},g ₃₄=−π₀ H ₀ ⁽²⁾(r ₂/√{square root over (λ₀)})/√{square root over(λ₀)},  (I.52)g ₄₁=((π₁+π₂ R ₃)λ₁ +iκb (1−R ₃))H ₀ ⁽¹⁾(r ₂/√{square root over(λ₁)})/√{square root over (λ₁)},  (I.53)g ₄₂=((π₁ R ₂+π₂)λ₂ +iκb (R ₂−1))H ₀ ⁽¹⁾(r ₂/√{square root over(λ₂)})/√{square root over (λ₂)},  (I.54)g ₄₃=−(ρ⁽¹⁾/ρ⁽⁰⁾)π₀ H ₀ ⁽¹⁾(r ₂/√{square root over (λ₀)})/√{square rootover (λ₀)},  (I.55)g ₄₄=−(ρ⁽¹⁾/ρ⁽⁰⁾)π₀ H ₀ ⁽²⁾(r ₂/√{square root over (λ₀)})/√{square rootover (λ₀)},  (I.56)

The solution to set (I.44) yields integration constants B₁, B₂, B₀, C₀:Ξ=G ⁻¹ P.  (I.57)It should be noted that pressure is in the units of the elastic modulusμ=ρ_(s)c_(t) ², and velocities u, v are in the units of the shear soundvelocity c_(t).

APPENDIX II

The theory describing propagation of the guided Stoneley waves at theinterface of the fluid and the deformed saturated porous medium is, asin Appendix I, based on a linearized version of the filtration theory.

The acoustic and electrical fields in fluid are described by thefollowing linear equation:

$\begin{matrix}{{{\overset{¨}{v} - {c_{p\; 0}^{2}{\nabla{div}}\; v}} = 0},} & \left( {{II}{.1}} \right) \\{{{\overset{.}{B} - {\frac{c_{e}^{2}}{4\pi\;\sigma}\Delta\; B}} = 0},} & \left( {{II}{.2}} \right) \\{E = {\frac{c_{e}}{4{\pi\sigma}}{rot}\;{B.}}} & \left( {{II}{.3}} \right)\end{matrix}$The acoustic and electrical fields in the saturated porous medium aredescribed by the following set of linear equations:

$\begin{matrix}{{{\overset{¨}{u} - {c_{t}^{2}\Delta\; u} - {a_{1}{\nabla{div}}\; u} + {a_{2}{\nabla{div}}\; v} + {\frac{\rho_{l}}{\rho_{s}}{b_{\alpha}\left( {\overset{.}{u} - \overset{.}{v}} \right)}} + {\frac{\rho_{l}}{\rho_{s}}b_{e}{rot}\; B}} = 0},} & \left( {{II}{.4}} \right) \\{\mspace{20mu}{{{\overset{¨}{v} + {a_{3}{\nabla{div}}\; u} - {a_{4}{\nabla{div}}\; v} - {b_{\alpha}\left( {\overset{.}{u} - \overset{.}{v}} \right)} - {b_{e}{rot}\; B}} = 0},}} & \left( {{II}{.5}} \right) \\{\mspace{20mu}{{{\overset{.}{B} - {\frac{c_{e}^{2}}{4{\pi\sigma}}\Delta\; B} - {\frac{\alpha\; c_{e}\rho_{l}}{\sigma}{{rot}\left( {u - v} \right)}}} = 0},}} & \left( {{II}{.6}} \right) \\{\mspace{20mu}{E = {{\frac{c_{e}}{4{\pi\sigma}}{rot}\; B} - {\frac{{\alpha\rho}_{l}}{\sigma}{\left( {u - v} \right).}}}}} & \left( {{II}{.7}} \right)\end{matrix}$

Here ρ_(l), ρ_(s), are partial densities of the saturating fluid and theporous matrix, respectively, ρ=ρ_(l)+ρ_(s) is density of the saturatedporous medium, u is the velocity of the porous matrix, v is the velocityof the saturating fluid or the borehole fluid, c_(p0) is the velocity ofsound in the borehole fluid. The dissipative coefficients areb_(α)=ρ_(l)χ(1−α²/(σχ)), χ=η/(kρ_(l)ρ), b_(e)=αc_(e)/(4πσ), where η isdynamic viscosity the saturating fluid, k is permeability, α is theelectroacoustic constant of the porous medium, σ is electricconductivity, c_(e) is the velocity of light. Coefficients α_(j) in theequations above are determined by three elastic moduli λ, μ, γ:

$\begin{matrix}{{a_{1} = {\frac{1}{\rho_{s}}\left( {{\frac{\rho_{s}^{2}}{\rho^{2}}\gamma} + {\frac{\rho_{l}^{2}}{\rho^{2}}K} + {\frac{1}{3}\mu}} \right)}},{a_{2} = {\frac{\rho_{l}}{\rho_{s}}\left( {{\frac{\rho_{l}}{\rho^{2}}K} - {\frac{\rho_{s}}{\rho^{2}}\gamma}} \right)}},} & \left( {{II}{.8}} \right) \\{{{a_{3} = {{\frac{\rho_{l}}{\rho^{2}}K} - {\frac{\rho_{s}}{\rho^{2}}\gamma}}},{a_{4} = {{\frac{\rho_{l}}{\rho^{2}}K} + \frac{\rho_{l}}{\rho^{2}}}}}{{{where}\mspace{14mu} K} = {\lambda + {\frac{2}{3}{\mu.}}}}} & \left( {{II}{.9}} \right)\end{matrix}$

Three elastic moduli λ, μ, and γ are found via the acoustic velocitiesof the saturated porous medium: the first compressional velocity c_(p1),the second compressional velocity c_(p2), shear velocity c_(t), measuredat high frequency:μ=ρ_(s) c _(t) ²,  (II.10)

$\begin{matrix}{{K = {\frac{1}{2}\frac{\rho_{s}}{\rho_{l}}\left( {{\rho\; c_{p\; 1}^{2}} + {\rho\; c_{p\; 2}^{2}} - {\frac{8}{3}\rho_{l}c_{t}^{2}} - \sqrt{\left( {{\rho\; c_{p\; 1}^{2}} - \rho_{p\; 2}^{2}} \right)^{2} - {\frac{64}{9}\rho_{s}\rho_{l}c_{t}^{4}}}} \right)}},} & \left( {{II}{.11}} \right) \\{\gamma = {\frac{1}{2}{\left( {{\rho\; c_{p\; 1}^{2}} + {\rho\; c_{p\; 2}^{2}} - {\frac{8}{3}\rho_{s}c_{t}^{2}} + \sqrt{\left( {{\rho\; c_{p\; 1}^{2}} - \rho_{p\; 2}^{2}} \right)^{2} - {\frac{64}{9}\rho_{s}\rho_{l}c_{t}^{4}}}} \right).}}} & \left( {{II}{.12}} \right)\end{matrix}$These are considered known quantities.

Let us present equations (II.1-II.7) in the dimensionless form, like inAppendix I:

in the borehole fluid:

$\begin{matrix}{{{{\lambda_{0}{\nabla{div}}\; v} + v} = 0},} & \left( {{II}{.13}} \right) \\{{{{{ig}\;\Delta\; B} - B} = 0},} & \left( {{II}{.14}} \right) \\{E = {\frac{c_{t}}{c_{e}}g\;{rot}\; B}} & \left( {{II}{.15}} \right)\end{matrix}$in the saturated porous medium:

$\begin{matrix}{{{{\Delta\; u} + {{\overset{\_}{a}}_{1}{\nabla{div}}\; u} - {{\overset{\_}{a}}_{2}{\nabla{div}}\; v} + {\left( {1 + {ɛ\;{\overset{\_}{b}}_{\alpha}}} \right)u} - {ɛ\;{\overset{\_}{b}}_{\alpha}v} + {ɛ\;{\overset{\_}{b}}_{e}{rot}\; B}} = 0},} & \left( {{II}{.16}} \right) \\{\mspace{20mu}{{{{{- {\overset{\_}{a}}_{3}}{\nabla{div}}\; u} + {{\overset{\_}{a}}_{4}{\nabla{div}}\; v} - {{\overset{\_}{b}}_{\alpha}u} + {\left( {1 + {\overset{\_}{b}}_{\alpha}} \right)v} - {{\overset{\_}{b}}_{e}{rot}\; B}} = 0},}} & \left( {{II}{.17}} \right) \\{\mspace{20mu}{{{{{ig}\;\Delta\; B} + {i\;{{rot}\left( {u - v} \right)}} - B} = 0},}} & \left( {{II}{.18}} \right) \\{\mspace{20mu}{E = {{\frac{c_{t}}{c_{e}}g\;{rot}\; B} - {\frac{c_{t}}{c_{e}}{\left( {u - v} \right).}}}}} & \left( {{II}{.19}} \right)\end{matrix}$Here, velocities u, v are in units of the shear velocity of sound c_(t);coordinates—in units of c_(t)/ω; B—in units of αc_(e)ρ_(l)/σ.Dimensionless coefficients in these equations are: λ₀=c_(p0) ²/c_(t) ²,ā_(i)=a_(i)/c_(t) ², b _(α)=ib_(α)/ω, ε=ρ_(l)/ρ_(s), g=ωc_(e) ²/(4πσc_(t) ²), b _(e)=iα²ρ_(l)c_(e) ²/(4 πσ²c_(t) ²).

To study dispersive properties of the Stoneley waves, one needs thedispersion ratio. To obtain it, let us introduce potentials which willenable us to separate compressional and shear waves in the medium underconsideration, with the help of the following equations:u=∇φ _(u)−rot ψ_(u),  (II.20)v=∇φ _(v)−rot ψ_(v).  (II.21)To study the Stoneley waves with the source in the center of thewellbore, it suffices to keep the cylindrical symmetry of the problem.The hydrodynamic velocities of the media should have componentsu=(u_(r), 0, u_(z)) and v=(v_(r), 0, v_(z)), while vector potentialshave only a single component ψ_(u)=(0, ψ_(u), 0) and ψ_(v)=(0, ψ_(v),0). The magnetic field should have component B=(0, B_(φ), 0).

In cylindrical coordinates, equations (II.20, II.21) have the followingform:

$\begin{matrix}{{u_{r} = {{\partial_{r}\varphi_{u}} + {\partial_{z}\psi_{u}}}},{u_{z} = {{\partial_{z}\varphi_{u}} - {\partial_{r}\psi_{u}} - {\frac{1}{r}\psi_{u}}}},} & \left( {{II}{.22}} \right) \\{{v_{r} = {{\partial_{r}\varphi_{v}} + {\partial_{z}\psi_{v}}}},{v_{z} = {{\partial_{z}\varphi_{v}} - {\partial_{r}\psi_{v}} - {\frac{1}{r}{\psi_{v}.}}}}} & \left( {{II}{.23}} \right)\end{matrix}$Two independent sub-systems describing compressional and shear waves inthe saturated porous medium are obtained via substituting (II.22, II.23)in (II.16-II.18):(ā ₁+1)Δφ_(u) −ā ₂Δφ_(v)+(1+εb _(α))φ_(u) −εb _(α)φ_(v)=0,  (II.24)−ā ₃Δφ_(u) +ā ₄Δφ_(v) −b _(α)φ_(u)+(1+ b _(α))φ_(v)=0,  (II.25)Δ_(φ)ψ_(u)+(1+εb _(α))ψ_(u) −εb _(α)ψ_(v) −εb _(e) B _(φ)=0,  (II.26)− b _(α)ψ_(u)+(1+b _(α))ψ_(v) +b _(e) B _(φ)=0,  (II.27)igΔ _(φ) B _(φ) +iΔ _(φ)ψ_(u) −iΔ _(φ)ψ_(v) −B _(φ)=0.  (II.28)

Here we define the operators

${{\Delta\;\varphi} = {{\partial_{r}^{2}\varphi} + {\frac{1}{r}{\partial_{r}\varphi}} + {{\partial_{z}^{2}\varphi}\mspace{14mu}{and}}}}\mspace{14mu}$${\Delta_{\varphi}\psi_{i}} = {{\partial_{r}^{2}\psi_{i}} + {\frac{1}{r}{\partial_{r}\psi_{i}}} - {\frac{1}{r^{2}}\psi_{i}} + {{\partial_{z}^{2}\psi_{i}}.}}$Along the z axis, for the symmetry assumed, the Stoneley wave is a planewave:(φ_(u),φ_(v),ψ_(u),ψ_(v) ,B)→(φ_(u),φ_(v),ψ_(u),ψ_(v),B)·exp(ikz−iωt)  (II.29)Let us find common solutions to equations (II.1, II.2, II.4-II.6).

Let us consider set of equations (II.24, II.25) determining theevolution of the compressional waves in the saturated porous medium:(ā ₁+1)Δφ_(u) −ā ₂Δφ_(v)+(1+εb _(α))φ_(u) −εb _(α)φ_(v)=0,  (II.30)−ā ₃Δφ_(u) +ā ₄Δφ_(v) −b _(α)φ_(u)+(1+b _(α))φ_(v)=0,  (II.31)Taking into account the form of the solution, set of equations(II.30-II.31) determining the evolution of the compressional waves maybe written out in the matrix form:AΔΦ+WΦ=0.  (II.32)Here

$A = \begin{pmatrix}{\overset{\_}{a}}_{0} & {- {\overset{\_}{a}}_{2}} \\{- {\overset{\_}{a}}_{3}} & {\overset{\_}{a}}_{4}\end{pmatrix}$is the matrix of coefficients,

${{\overset{\_}{a}}_{0} = {{\overset{\_}{a}}_{1} + 1}},{W = \begin{pmatrix}\left( {1 + {ɛ{\overset{\_}{b}}_{a}}} \right) & {{- ɛ}{\overset{\_}{b}}_{a}} \\{- {\overset{\_}{b}}_{a}} & \left( {1 + {\overset{\_}{b}}_{a}} \right)\end{pmatrix}},$whereas the other coefficients are defined above; Φ=(φ_(u) φ_(v))^(T) isthe vector-column of functions.

We transform set of equations (II.32) via multiplying it by the matrixW⁻¹:

${W^{- 1} = \begin{pmatrix}\frac{1 + {\overset{\_}{b}}_{a}}{\det\; W} & \frac{ɛ{\overset{\_}{b}}_{a}}{\det\; W} \\\frac{{\overset{\_}{b}}_{a}}{\det\; W} & \frac{1 + {ɛ{\overset{\_}{b}}_{a}}}{\det\; W}\end{pmatrix}},$where det W=1+(1+ε)b _(α), and we arrive at the set of equations withthe diagonal matrix in front of the free term:HΔΦ+Φ=0, where H=W ⁻¹ A.  (II.33)The components of the matrix H are:h ₁₁=((1+ b )ā ₀ −εbā ₃)/det W, h ₁₂=(−(1+b )ā ₂ +εbā ₄)/det W,  (II.34)h ₂₁=( bā ₀−(1+ε b )ā ₃)/det W, h ₂₂=(− bā ₂+(1+εb )ā ₄)/det W.  (II.35)Let us bring set of equations (II.33) obtained above to the diagonalform:

$\begin{matrix}{{{{\Lambda\Delta\Omega} + \Omega} = 0},{where}} & \left( {{II}{.36}} \right) \\{{\Lambda = \begin{pmatrix}\lambda_{1} & 0 \\0 & \lambda_{2}\end{pmatrix}},{\Omega = \left( {\varphi_{1}\mspace{14mu}\varphi_{2}} \right)^{T}}} & \left( {{II}{.37}} \right)\end{matrix}$using the transform matrix R

$\begin{matrix}{{R = \left( {\begin{pmatrix}R_{1} \\R_{3}\end{pmatrix}\begin{pmatrix}R_{2} \\R_{4}\end{pmatrix}} \right)},} & \left( {{II}{.38}} \right)\end{matrix}$which is made of the right-hand side eigenvectors of the matrix A:R⁻¹AR=Λ. HereR ₁=1, R ₂=(λ₂ −h ₂₂)/h ₂₁ , R ₃ =h ₂₁/(λ₁ −h ₂₂), R ₄=1.  (II.39)

The column vector of these functions is transformed as follows:

$\begin{matrix}{{{R^{- 1}\Phi} = \Omega},{{{where}\mspace{14mu}\Omega} = \begin{pmatrix}\varphi_{1} \\\varphi_{2}\end{pmatrix}}} & \left( {{II}{.40}} \right)\end{matrix}$The diagonal matrix Λ=R⁻¹AR has eigenvalues λ₁ and λ₂ of the matrix A asits diagonal elements, which are determined as follows:λ₁=½tr H−√{square root over (¼(tr H)²−det H)}, λ₂=½tr H+√{square rootover (¼(tr H)²−det H)},  (II.41)where tr H=h₁₁+h₂₂ is the trace of the matrix H, det H=h₁₁h₂₂−h₁₂h₂₁ isthe determinant of the matrix H.

Thus, set of equations (II.30, II.31) in cylindrical coordinates isexpressed in the diagonal form, (II.29)

$\begin{matrix}{{{{\partial_{r}^{2}\varphi_{1}} + {\frac{1}{r}{\partial_{r}\varphi_{1}}} + {\frac{1}{\lambda_{1}}\varphi_{1}} + {\partial_{z}^{2}\varphi_{1}}} = 0},} & \left( {{II}{.42}} \right) \\{{{{\partial_{r}^{2}\varphi_{2}} + {\frac{1}{r}{\partial_{r}\varphi_{2}}} + {\frac{1}{\lambda_{2}}\varphi_{2}} + {\partial_{z}^{2}\varphi_{2}}} = 0},} & \left( {{II}{.43}} \right)\end{matrix}$or, taking into account the form of solution (II.29),

$\begin{matrix}{{{\varphi_{1}^{''} + {\frac{1}{r}\varphi_{1}^{\prime}} - {l_{1}^{2}\varphi_{1}}} = 0},} & \left( {{II}{.44}} \right) \\{{{\varphi_{2}^{''} + {\frac{1}{r}\varphi_{2}^{\prime}} - {l_{2}^{2}\varphi_{2}}} = 0},} & \left( {{II}{.45}} \right)\end{matrix}$where it is determined that

${l_{1}^{2} = {k^{2} - \frac{1}{\lambda_{1}}}},{l_{2}^{2} = {k^{2} - {\frac{1}{\lambda_{2}}.}}}$The prime sign marks the derivative with respect to radius.

Solutions to equations (II.44, II.45) are expressed via the modifiedBessel functions K₀(l_(i)r), I₀(l_(i)r) and, taking into account theboundedness of this solution in the infinity, they have the followingform:φ₁ =C ₁ K ₀(l ₁ r), φ₂ =C ₂ K ₀(l ₂ r)  (II.46)Here C₁, C₂ are integration constants.The solution to set (II.44, II.45) and the solution to the original set(II.30, II.31) are related as in (II.40):Φ=RΩ  (II.47)Orφ_(u)=φ₁ +R ₂φ₂,  (II.48)φ_(v) =R ₃φ₁+φ₂.  (II.49)Thus, the solution for the potentials φ_(u), φ_(v) can be written out asfollows:φ_(u) =C ₁ K ₀(l ₁ r)+C ₂ R ₂ K ₀(l ₂ r), φ_(v) =C ₁ R ₃ K ₀(l ₁ r)+C ₂K ₀(l ₂ r).   (II.50)

Let us next consider the set of equations (II.26-II.28) determining theevolution of the shear waves:Δ_(φ)ψ_(u)+(1+εb _(α))ψ_(u) −εb _(α)ψ_(v) −εb _(e) B _(φ)=0,  (II.51)− b _(α)ψ_(u)+(1+b _(α))ψ_(v) +b _(e) B _(φ)=0,  (II.52)igΔ _(φ) B _(φ) +iΔ _(φ)ψ_(u) −iΔ _(φ)ψ_(v) −B _(φ)=0.  (II.53)The second equation yields the following relationship between thepotentials:

$\begin{matrix}{{\psi_{v} = {{{\overset{\_}{b}}_{a}{\beta\psi}_{u}} - {{\overset{\_}{b}}_{e}\beta\; B_{\varphi}}}},{{{where}\mspace{14mu}\beta} = \frac{1}{1 + {\overset{\_}{b}}_{a}}},} & \left( {{II}{.54}} \right)\end{matrix}$which allows us to exclude ψ_(v) from set (II.52):Δ_(φ)ψ_(u) +εb _(α)βψ_(u) −εb _(e) βB _(φ)=0,  (II.55)iβΔ _(φ)ψ_(u) +i(g+b _(e)β)Δ_(φ) B _(φ) −B _(φ)=0.  (II.56)

Taking into account the form of the solution, set of equations (II.55,II.56) determining the evolution of the compressional waves may bewritten out in the matrix form:DΔ _(φ) Ψ+WΨ=0.  (II.57)Here

$D = \begin{pmatrix}1 & 0 \\{i\;\beta} & {i\left( {g + {{\overset{\_}{b}}_{e}\beta}} \right)}\end{pmatrix}$is the matrix of coefficients,

${W = \begin{pmatrix}{ɛ{\overset{\_}{b}}_{a}\beta} & {{- ɛ}{\overset{\_}{b}}_{e}\beta} \\0 & {- 1}\end{pmatrix}},$while the other coefficients are defined above; Ψ=(ψ_(u) B_(φ))^(T) isthe vector-column of functions.We transform set of equations (II.57) via multiplying it by the matrixW⁻¹:

${W^{- 1} = \begin{pmatrix}\frac{- 1}{\det\; W} & \frac{ɛ{\overset{\_}{b}}_{e}\beta}{\det\; W} \\0 & \frac{ɛ{\overset{\_}{b}}_{\alpha}\beta}{\det\; W}\end{pmatrix}},$where det W=−εb _(e)β, and we arrive at the set of equations with thediagonal matrix in front of the free term:NΔ _(φ)Ψ+Ψ=0, where N=W ⁻¹ D.  (II.58)The components of the matrix N are:n ₁₁=(−1+εb _(e)β²)/det W, n ₁₂ =iεb _(e)β(g+b _(e)β)/det W,  (II.59)n ₂₁ =iεb _(α)β²/det W, n ₂₂ =iεb _(α)β(g+b _(e)β)/det W.  (II.60)

Let us bring set of equations (II.58) obtained above to the diagonalform:

$\begin{matrix}{{{{{\Lambda\Delta}_{\varphi}\Omega} + \Omega} = 0},{where}} & \left( {{II}{.61}} \right) \\{{\Lambda = \begin{pmatrix}\lambda_{3} & 0 \\0 & \lambda_{4}\end{pmatrix}},{\Omega = \left( {\psi_{1}\mspace{14mu}\psi_{2}} \right)^{T}}} & \left( {{II}{.62}} \right)\end{matrix}$using the transform matrix P

$\begin{matrix}{{P = \begin{pmatrix}P_{1} & P_{2} \\P_{3} & P_{4}\end{pmatrix}},} & \left( {{II}{.63}} \right)\end{matrix}$which is made of the right-hand side eigenvectors of the matrix D:P⁻¹DP=Λ. HereP ₁=1, P ₂=(λ₄ −n ₂₂)/n ₂₁ , P ₃ =n ₂₁/(λ₃ −n ₂₂), P ₄=1.  (II.64)The column vector of these functions is transformed as follows:

$\begin{matrix}{{{P^{- 1}\Psi} = \Omega},{{{where}\mspace{14mu}\Omega} = {\begin{pmatrix}\psi_{1} \\\psi_{2}\end{pmatrix}.}}} & \left( {{II}{.65}} \right)\end{matrix}$

The diagonal matrix Λ=P⁻¹DP has eigenvalues λ₃ and λ₄ of the matrix A asits diagonal elements, which are determined as follows:λ₃=½tr N−√{square root over (¼(tr N)²−det N)}, λ₄=½tr N+√{square rootover (¼(tr N)²−detN)},  (II.66)where tr N=n₁₁+n₂₂, det N=n₁₁n₂₂−n₁₂n₂₁.Thus, set of equations (II.55, II.56) in cylindrical coordinates isexpressed in the diagonal form, (II.61)

$\begin{matrix}{{{{\partial_{r}^{2}\psi_{1}} + {\frac{1}{r}{\partial_{r}\psi_{1}}} - {\frac{1}{r^{2}}\psi_{1}} + {\frac{1}{\lambda_{3}}\psi_{1}} + {\partial_{z}^{2}\psi_{1}}} = 0},} & \left( {{II}{.67}} \right) \\{{{{\partial_{r}^{2}\psi_{2}} + {\frac{1}{r}{\partial_{r}\psi_{2}}} - {\frac{1}{r^{2}}\psi_{2}} + {\frac{1}{\lambda_{4}}\psi_{2}} + {\partial_{z}^{2}\psi_{2}}} = 0},} & \left( {{II}{.68}} \right)\end{matrix}$or, taking into account the form of its solution (II.29), set ofequations (II.67, II.68) determining the evolution of the shear wavescan be reduced to the following set of equations:

$\begin{matrix}{{{\psi_{1}^{''} + {\frac{1}{r}\psi_{1}^{\prime}} - {\left( {l_{3}^{2} + \frac{1}{r^{2}}} \right)\psi_{1}}} = 0},} & \left( {{II}{.69}} \right) \\{{{\psi_{2}^{''} + {\frac{1}{r}\psi_{2}^{\prime}} - {\left( {l_{4}^{2} + \frac{1}{r^{2}}} \right)\psi_{2}}} = 0},} & \left( {{II}{.70}} \right)\end{matrix}$where it is determined that

${l_{3}^{2} = {k^{2} - \frac{1}{\lambda_{3}}}},{l_{4}^{2} = {k^{2} - {\frac{1}{\lambda_{4}}.}}}$The prime sign marks the derivative with respect to radius.Solutions to equations (II.67, II.68) are expressed via the modifiedBessel functions K₁(l_(i)r), I₁(l_(i)r) and, taking into account theboundedness of this solution in the infinity, they have the followingform:ψ₁ =C ₃ K ₁(l ₃ r), ψ₂ =C ₄ K ₁(l ₄ r).  ((II.71)Here C₃, C₄ are integration constants.The solution to set (II.67, II.68) and the solution to the original set(II.55, II.56) are related as in (II.65):Ψ=PΩ  (II.72)Orψ_(u)=ψ₁ +P ₂ψ₂,  (II.73)B _(φ) =P ₃ψ₁+ψ₂.  (II.74)

The solution for the potentials ψ_(u), ψ_(v) and magnetic field B_(φ)are expressed via the solution for the potentials ψ₁, ψ₂:ψ_(u) =C ₃ K ₁(l ₃ r)+C ₄ P ₂ K ₁(l ₄ r),  (II.75)B _(φ) =C ₃ P ₃ K ₁(l ₃ r)+C ₄ K ₁(l ₄ r),  (II.76)ψ_(v) =C ₃β₃ K ₀(l ₃ r)+C ₄β₄ K ₀(l ₄ r),  (II.77)where β₃=( b _(α) −b _(e) P ₃)β, β₄=( b _(α) P ₂ −b _(e))β.

Solutions (II.75, I.76, I.77) obtained above lead to expressions forvelocities and time derivatives of the stress tensors. Due to the choiceof geometry in this problem, velocity components can be expressed asfollows:

$\begin{matrix}\begin{matrix}{{u_{r} = {{\partial_{r}\varphi_{u}} + {\partial_{z}\psi_{u}}}},} & {{u_{z} = {{\partial_{z}\varphi_{u}} - {\partial_{r}\psi_{u}} - {\frac{1}{r}\psi_{u}}}},}\end{matrix} & \left( {{II}{.78}} \right) \\\begin{matrix}{{v_{r} = {{\partial_{r}\varphi_{v}} + {\partial_{z}\psi_{v}}}},} & {v_{z} = {{\partial_{z}\varphi_{v}} - {\partial_{r}\psi_{v}} - {\frac{1}{r}{\psi_{v}.}}}}\end{matrix} & \left( {{II}{.79}} \right)\end{matrix}$Because, from (II.48, II.49, II.73, II.74):φ_(u)=φ₁ +R ₂φ₂, φ_(v) =R ₃φ₁+φ₂,  (II.80)ψ_(u)=ψ₁ +P ₂ψ₂ , B _(φ) =P ₃ψ₁+ψ₂, ψ_(v)=β₃ψ₁+β₄ψ₂,  (II.81)we have

$\begin{matrix}{\mspace{79mu}{{u_{r} = {{\partial_{r}\varphi_{1}} + {R_{2}{\partial_{r}\varphi_{2}}} + {\partial_{z}\psi_{1}} + {P_{2}{\partial_{z}\psi_{2}}}}},}} & \left( {{II}{.82}} \right) \\{\mspace{79mu}{{u_{z} = {{\partial_{z}\varphi_{1}} + {R_{2}{\partial_{z}\varphi_{2}}} - {\partial_{r}\psi_{1}} - {P_{2}{\partial_{r}\psi_{2}}} - {\frac{1}{r}\psi_{1}} - {\frac{1}{r}P_{2}\psi_{2}}}},}} & \left( {{II}{.83}} \right) \\{\mspace{79mu}{{v_{r} = {{R_{3}{\partial_{r}\varphi_{1}}} + {\partial_{r}\varphi_{2}} + {\beta_{3}{\partial_{z}\psi_{u}}} + {\beta_{4}{\partial_{z}\beta_{\varphi}}}}},}} & \left( {{II}{.84}} \right) \\{v_{z} = {{R_{3}{\partial_{z}\varphi_{1}}} + {\partial_{z}\varphi_{2}} - {\beta_{3}{\partial_{r}\psi_{v}}} - {\beta_{4}{\partial_{r}\beta_{\varphi}}} - {\frac{1}{r}\beta_{3}\psi_{u}} - {\frac{1}{r_{e}}\beta_{4}{\beta_{\varphi}.}}}} & \left( {{II}{.85}} \right)\end{matrix}$

In accordance to (II.50, II.75-I.77) and taking into account therecurrent relationships between the Bessel functions, we arrive at thefinal form of the solution to set of equations (II.16-II.18):u _(r) =−C ₁ l ₁ K ₁(l ₁ r)−C ₂ R ₂ l ₂ K ₁(l ₂ r)+ikC ₃ K ₁(l ₃ r)+ikC₄ P ₂ K ₁(l ₄ r),  (II.86)u _(z) =ikC ₁ K ₀(l ₁ r)+ikC ₂ R ₂ K ₀(l ₂ r)+C ₃ l ₃ K ₀(l ₃ r)+C ₄ P ₂l ₄ K ₀(l ₄ r),  (II.87)v _(r) =−C ₁ R ₃ l ₁ K ₁(l ₁ r)−C ₂ l ₂ K ₁(l ₂ r)+ikC ₃β₃ K ₁(l ₃r)+ikC ₄β₄ K ₁(l ₄ r),  (II.88)v _(z) =ikC ₁ R ₃ K ₀(l ₁ r)+ikC ₂ K ₀(l ₂ r)+C ₃β₃ l ₄ K ₀(l ₄r),  (II.89)B _(φ) =C ₃ P ₃ K ₁(l ₃ r)+C ₄ K ₁(l ₄ r)  (II.90)

Let us next consider the equation determining the evolution of theborehole fluid. We can introduce potentials for the fluid velocity justas in the discussion above, via the following expression:v=∇φ _(f)−rot ψ_(f),  (II.91)which has the following components in the given geometry:v _(r)=∂_(r)φ_(f),  (II.92)v _(z)=∂_(z)φ_(f).  (II.93)Then, in accordance to equation (II.13), we arrive at:λ₀Δφ_(f)+φ_(f)=0.  (II.94)

The solution to equation (II.94) for the borehole fluid is expressed viathe modified Bessel functions I_(α)(z):

$\begin{matrix}{{\varphi_{v} = {C_{5}{I_{0}\left( {l_{0}r} \right)}}},{l_{0}^{2} = {k^{2} - {\frac{1}{\lambda_{0}}.}}}} & \left( {{II}{.95}} \right)\end{matrix}$Here we take into account the fact that no singularity is found in thecenter of the borehole.

Let us next consider the equation determining the evolution of magneticfield in the borehole (II.14):igΔ _(φ) B _(φ) −B _(φ)=0,  (II.96)The solution to equation (II.96) for the borehole fluid is expressed viathe modified Bessel functions I_(α)(z):

$\begin{matrix}{{B_{\varphi} = {C_{6}{I_{1}\left( {l_{6}r} \right)}}},{l_{6}^{2} - {\frac{i}{g}.}}} & \left( {{II}{.97}} \right)\end{matrix}$Here we take into account the fact that no singularity is found in thecenter of the borehole. The solutions to equations (II.13, II.14) forthe borehole are:v _(r) =C ₅ l ₀ I ₁(l ₀ r),  (II.98)v _(z) =ikC ₅ I ₀(l ₀ r),  (II.99)B _(φ) =C ₆ I ₁(l ₆ r).  (II.100)We arrive at the dispersive relationship for the wave vector k inaccordance to the condition of solutions (II.86-II.90, II.98-II.100)satisfying the boundary conditions at the borehole boundary:

The continuity condition for the φ component of the magnetic field and zcomponent of the electrical field:B _(φ) ⁽¹⁾ =B _(φ) ⁽⁰⁾  (II.101)E _(z) ⁽¹⁾ =E _(z) ⁽⁰⁾  (II.102)The continuity condition for the normal mass flow:(1−φ)u _(r) ⁽¹⁾ +φv _(r) ⁽¹⁾ =v _(r) ⁽⁰⁾  (II.103)The continuity condition for the components of the normal projection ofthe stress tensor Σ_(ik)Σ_(rr) ⁽¹⁾=Σ_(rr) ⁽⁰⁾,  (II.104)Σ_(rz) ⁽¹⁾=0,  (II.105)

The equality condition for the components of the normal projection ofthe partial stress tensor of the saturating fluid Σ_(rr) ^(l(1)) and theborehole fluid Σ_(rr) ⁽⁰⁾Σ_(rr) ^(l(1))=φΣ_(rr) ⁽⁰⁾.  (II.106)The latter condition in terms of pressure takes the following form:

$\begin{matrix}{{\frac{p^{(1)}}{\rho^{(1)}} - {\kappa\frac{\rho^{(1)}}{\rho_{s}^{(1)}}{\overset{\_}{b}\left( {\varphi_{u}^{(1)} - \varphi_{v}^{(1)}} \right)}}} = {\frac{p^{(0)}}{\rho^{(0)}}.}} & \left( {{II}{.107}} \right)\end{matrix}$Here the coefficient κ is introduced because there is no reason tobelieve that the friction coefficient at the boundary has to coincidewith the friction coefficient in the porous medium. Consequently, thecontinuity condition is satisfied for the time derivatives of thecomponents of the normal projection of the elastic tensor:{dot over (Σ)}_(rz) ⁽¹⁾=0, {dot over (Σ)}_(rr) ⁽¹⁾={dot over (Σ)}_(rr)⁽⁰⁾, {dot over (Σ)}_(rr) ⁽¹⁾=φ{dot over (Σ)}_(rr) ⁽⁰⁾.  (II.108)Here−{dot over (p)} ⁽¹⁾=π₁ div u ⁽¹⁾+π₂ div v ⁽¹⁾,  (II.109){dot over (Σ)}_(rr) ⁽¹⁾=π₃ div u ⁽¹⁾+π₄ div v ⁽¹⁾+2∂_(r) u_(r),  (II.110){dot over (Σ)}_(rz) ⁽¹⁾=∂_(r) u _(z)+∂_(z) u _(r),  (II.111)

$\begin{matrix}{{{\overset{.}{\Sigma}}_{rr}^{(0)} = {{- {\overset{.}{p}}^{(0)}} = {\pi_{0}{divv}^{(0)}}}},} & \left( {{II}{.112}} \right) \\{{\pi_{1} = {\left( {{\frac{\rho_{s}}{\rho}\gamma} - {\frac{\rho_{l}}{\rho}K}} \right)/\left( {\rho_{s}c_{t}^{2}} \right)}},{{\pi_{2}\left( {{\frac{\rho_{l}}{\rho}\gamma} + {\frac{\rho_{l}}{\rho}K}} \right)}/\left( {\rho_{s}c_{t}^{2}} \right)},} & \left( {{II}{.113}} \right) \\{{\pi_{3} = {\left( {{\frac{\rho_{s}}{\rho}\gamma} - {\frac{2}{3}\mu}} \right)/\left( {\rho_{s}c_{t}^{2}} \right)}},{{\pi_{4}\left( {\frac{\rho_{l}}{\rho}\gamma} \right)}/\left( {\rho_{s}c_{t}^{2}} \right)},} & \left( {{II}{.114}} \right) \\{\pi_{0} = {\rho^{(0)}{c_{p\; 0}^{2}/{\left( {\rho_{s}c_{t}^{2}} \right).}}}} & \left( {{II}{.115}} \right)\end{matrix}$

Substituting equations (II.93-II.100) into the boundary conditions(II.101-II.106), we can then present the boundary conditions in thematrix form:GΞ=0.  (II.116)where the vector-column of coefficients Ξ=(C₁ C₂ . . . C₅ C₆)^(T) andthe matrix G with the components

$\begin{matrix}{\mspace{79mu}{{g_{11} = {\left( {\left( {1 - \phi} \right) + {R_{3}\phi}} \right)l_{1}{K_{1}\left( {l_{1}r_{2}} \right)}}},\mspace{79mu}{g_{12} = {\left( {{\left( {1 - \phi} \right)R_{2}} + \phi} \right)l_{2}{K_{1}\left( {l_{2}r_{2}} \right)}}},}} & \left( {{II}{.117}} \right) \\{\mspace{79mu}{{g_{13} = {{- {\mathbb{i}}}\;{k\left( {\left( {1 - \phi} \right) + {\beta_{3}\phi}} \right)}{K_{1}\left( {l_{3}r_{2}} \right)}}},\mspace{79mu}{g_{14} = {{- {\mathbb{i}}}\;{k\left( {{\beta_{4}\phi} - {\left( {1 - \phi} \right)P_{2}}} \right)}{K_{1}\left( {l_{3}r_{2}} \right)}}},}} & \left( {{II}{.118}} \right) \\{\mspace{79mu}{{g_{15} = {{- l_{0}}{l_{1}\left( {l_{0},r_{2}} \right)}}},{g_{16} = 0},}} & \left( {{II}{.119}} \right) \\{{g_{21} = {{\left( {{\left( {\pi_{3} + {\pi_{4}R_{3}}} \right)\left( {l_{1}^{2} - k^{2}} \right)} + {2\; l_{1}^{2}}} \right){K_{0}\left( {l_{1}r_{2}} \right)}} + {2\; l_{1}{{K_{1}\left( {l_{1}r_{2}} \right)}/r_{2}}}}},} & \left( {{II}{.120}} \right) \\{{g_{22} = {{\left( {{\left( {{\pi_{3}R_{2}} + \pi_{4}} \right)\left( {l_{2}^{2} - k^{2}} \right)} + {2\; R_{2}l_{2}^{2}}} \right){K_{0}\left( {l_{2}r_{2}} \right)}} + {2\; R_{2}l_{2}{{K_{2}\left( {l_{2}r_{2}} \right)}/r_{2}}}}},} & \left( {{II}{.121}} \right) \\{\mspace{79mu}{{g_{23} = {{- 2}{\mathbb{i}}\;{k\left( {{l_{3}{K_{0}\left( {l_{3}r_{2}} \right)}} + {{K_{1}\left( {l_{3}r_{2}} \right)}/r_{2}}} \right)}}},}} & \left( {{II}{.122}} \right) \\{\mspace{79mu}{{g_{24} = {{- 2}\;{\mathbb{i}}\;{{kP}_{2}\left( {{l_{4}{K_{0}\left( {l_{4}r_{2}} \right)}} + {{K_{1}\left( {l_{4}r_{2}} \right)}/r_{2}}} \right)}}},}} & \left( {{II}{.123}} \right) \\{\mspace{79mu}{{g_{25} = {{- {\pi_{0}\left( {l_{0}^{2} - k^{2}} \right)}}{I_{0}\left( {l_{0}r_{2}} \right)}}},{g_{26} = 0},}} & \left( {{II}{.124}} \right) \\{\mspace{79mu}{{g_{31} = {2\;{\mathbb{i}}\;{kl}_{1}{K_{1}\left( {l_{1}r_{2}} \right)}}},{g_{32} = {2\;{\mathbb{i}}\;{kR}_{2}l_{2}{K_{1}\left( {l_{2}r_{2}} \right)}}},}} & \left( {{II}{.125}} \right) \\{\mspace{79mu}{{g_{33} = {\left( {l_{3}^{2} + k^{2}} \right){K_{1}\left( {l_{3}r_{2}} \right)}}},{g_{34} = {{P_{2}\left( {l_{3}^{2} + k^{2}} \right)}{K_{1}\left( {l_{3}r_{2}} \right)}}},}} & \left( {{II}{.126}} \right) \\{\mspace{79mu}{{g_{35} = 0},{g_{36} = 0},}} & \left( {{II}{.127}} \right) \\{\mspace{79mu}{{g_{41} = {\left( {{\left( {\pi_{1} + {R_{3}\pi_{2}}} \right)\left( {l_{1}^{2} - k^{2}} \right)} - {{\overset{\_}{b}}_{\alpha}\frac{\rho^{(1)}}{\rho_{s}^{(1)}}\left( {1 - R_{3}} \right)}} \right){K_{0}\left( {l_{1}r_{2}} \right)}}},}} & \left( {{II}{.128}} \right) \\{\mspace{79mu}{{g_{42} = {\left( {{\left( {{R_{2}\pi_{1}} + \pi_{2}} \right)\left( {l_{2}^{2} - k^{2}} \right)} - {{\overset{\_}{b}}_{\alpha}\frac{\rho^{(1)}}{\rho_{s}^{(1)}}\left( {R_{2} - 1} \right)}} \right){K_{0}\left( {l_{2}r_{2}} \right)}}},}} & \left( {{II}{.129}} \right) \\{\mspace{79mu}{{g_{43} = 0},{g_{44} = 0},{g_{45} = {{- \frac{\rho^{(1)}}{\rho^{(0)}}}{\pi_{0}\left( {l_{0}^{2} - k^{2}} \right)}{I_{0}\left( {l_{0}r_{2}} \right)}}},{g_{46} = 0},}} & \left( {{II}{.130}} \right) \\{\mspace{79mu}{{g_{51} = {{\mathbb{i}}\;{k\left( {1 - R_{3}} \right)}{K_{0}\left( {l_{1}r_{2}} \right)}}},{g_{52} = {{\mathbb{i}}\;{k\left( {R_{2} - 1} \right)}{K_{0}\left( {l_{2}r_{2}} \right)}}},}} & \left( {{II}{.131}} \right) \\{{g_{53} = {\left( {1 + {gP}_{3} - \beta_{3}} \right)l_{3}{K_{0}\left( {l_{3}r_{2}} \right)}}},{g_{54} = {\left( {P_{2} + g - \beta_{4}} \right)l_{4}{K_{0}\left( {l_{4}r_{2}} \right)}}},} & \left( {{II}{.132}} \right) \\{\mspace{79mu}{{g_{55} = 0},{g_{56} = {{gl}_{6}{I_{0}\left( {l_{6}r_{2}} \right)}}},}} & \left( {{II}{.133}} \right) \\{\mspace{79mu}{{g_{61} = 0},{g_{62} = 0},{g_{63} = {P_{3}{K_{1}\left( {l_{3}r_{2}} \right)}}},{g_{64} = {K_{1}\left( {l_{4}r_{2}} \right)}},}} & \left( {{II}{.134}} \right) \\{\mspace{79mu}{{g_{65} = 0},{g_{66} = {- {{I_{1}\left( {l_{6}r_{2}} \right)}.}}}}} & \left( {{II}{.135}} \right)\end{matrix}$

The dispersive relationship which ties together the wave vector k andfrequency ω, is found based on the fact that the matrix G determinant isnil:det G=0  (II.136)Finding roots to this non-linear equation is performed numerically usingthe Mueller method. The oscillatory mode k_(st) characterizing Stoneleywave is selected among the roots of a dispersive relationship. Thevelocity of the guided wave v_(st), the attenuation length along theboundary l_(*) and the attenuation coefficient 1/Q are determined by thefollowing equations:

$\begin{matrix}{{v_{st} = {\omega\left( {{Re}\; k_{st}} \right)}^{- 1}},{l_{*} = \left( {{Im}\; k_{st}} \right)^{- 1}},{\frac{1}{Q} = {2{\frac{{Im}\left( k_{st} \right)}{{Re}\left( k_{st} \right)}.}}}} & \left( {{II}{.137}} \right)\end{matrix}$In order to find the electroacoustic ratio (3) let us write out thesolution for the z-component of electrical field in borehole (II.15) indimensionless form, taking into account the solution for φ-component ofmagnetic field (II.100)

$\begin{matrix}{E_{z} = {C_{6}\frac{c_{l}}{c_{e}}{gl}_{6}{{I_{0}\left( {l_{6}r} \right)}.}}} & \left( {{II}{.138}} \right)\end{matrix}$where

$l_{6}^{2} = {k_{st}^{2} - {\frac{i}{g}.}}$The expression (II.138) and the solution for velocity of borehole fluid(II.99) lead to the electroacoustic ratio

$\begin{matrix}{{\frac{E_{z}}{v_{z}} = {{- i}\frac{C_{6}c_{l}g\sqrt{\left( {k_{st}^{2} - {i/g}} \right)}{I_{0}\left( {r\sqrt{\left( {k_{st}^{2} - {i/g}} \right)}} \right)}}{C_{5}c_{e}k_{st}{I_{0}\left( {r\sqrt{k_{st}^{2} - {1/\lambda_{0}}}} \right)}}}},} & \left( {{II}{.139}} \right)\end{matrix}$The constant of integration C₆/C₆ can be defined from set of equations(II.116).

The invention claimed is:
 1. A method of evaluating an earth formationintersected by a borehole, the method comprising: using an acoustictransmitter to generate acoustic signals at a frequency producing aresonance of fluid wherein said resonance generates a velocity of thefluid in the borehole and a velocity of the formation at an interfacebetween the fluid and the formation such that a difference between thevelocity of the fluid in the borehole and the velocity of the formationis dependent upon permeability of the formation; and making a firstmeasurement of the fluid velocity; making a second measurement of thevelocity of the formation; making a third measurement indicative of anelectric charge on a wall of the borehole taken at the frequency; andusing a processor to estimate an electroacoustic constant of the earthformation using the first measurement, the second measurement, and thethird measurement.
 2. The method of claim 1 wherein the acoustictransmitter comprises a swept frequency source, and wherein using theacoustic transmitter further comprises using the swept frequency sourcein the borehole and identifying the frequency by identifying from aplurality of frequencies in a frequency sweep the frequency at which acorresponding difference between a corresponding fluid velocity in theborehole and a corresponding velocity of the formation is at a maximum.3. The method of claim 2 further comprising using a geophone formeasuring the velocity of the formation and using a flow rate sensor tomeasure the fluid velocity.
 4. The method of claim 2 further comprising:making an additional first measurement of the fluid velocity and anadditional second measurement of the velocity of the formation at atleast one additional frequency different from the frequency; and usingthe additional first measurement and the additional second measurementto estimate a permeability of the earth formation.
 5. The method ofclaim 4 further comprising using a measured value of a firstcompressional wave in the formation, a second compressional waves in theformation, and a shear velocity of the formation to estimate thepermeability.
 6. The method of claim 2 wherein using the swept frequencysource further comprises using a monopole source.
 7. The method of claim2 further comprising conveying the swept frequency source into theborehole on a conveyance device selected from: (i) a wireline, and (ii)a drilling tubular.
 8. The method of claim 1 wherein making the thirdmeasurement indicative of the electric charge further comprises at leastone of (i) using a grounded ammeter, and (ii) making a measurement in apilot hole in the borehole wall.
 9. An apparatus configured to evaluatean earth formation intersected by a borehole, the apparatus comprising:a device including an acoustic transmitter configured to generateacoustic signals at a frequency producing a resonance of fluid whereinsaid resonance generates a velocity of the fluid in the borehole and avelocity of the formation at an interface between the fluid and theformation such that a difference between the velocity of the fluid inthe borehole and the velocity of the formation is dependent uponpermeability of the formation; a first sensor configured to make a firstmeasurement of the fluid velocity; a second sensor configured to make asecond measurement of the velocity of the formation; a third sensorconfigured to make a third measurement indicative of an electric chargeon a wall of the borehole taken at the frequency; and a processorconfigured to estimate an electroacoustic constant of the earthformation using the first measurement, the second measurement, and thethird measurement.
 10. The apparatus of claim 9 wherein the acoustictransmitter comprises a swept frequency source; and wherein theprocessor is further configured to identify the frequency by identifyingfrom a plurality of frequencies in a frequency sweep the frequency atwhich a corresponding difference between a corresponding fluid velocityin the borehole and a corresponding velocity of the formation is at amaximum.
 11. The apparatus of claim 10 further comprising a geophoneconfigured to measure the velocity of the formation and a flow ratesensor configured to measure the fluid velocity.
 12. The apparatus ofclaim 10 wherein using the swept frequency source further comprisesusing a monopole source.
 13. The apparatus of claim 10 furthercomprising a conveyance device configured to convey the swept frequencysource into the borehole, the conveyance device being selected from: (i)a wireline, and (ii) a drilling tubular.
 14. The apparatus of claim 9wherein the third sensor is selected from: (i) a grounded ammeter, and(ii) a sensor in a pilot hole in the borehole wall.
 15. The apparatus ofclaim 9 wherein the processor is further configured to estimate apermeability of the earth formation using an additional firstmeasurement of the fluid velocity and an additional second measurementof the velocity of the formation at at least one additional frequency ofthe fluid different from the frequency.
 16. The apparatus of claim 15wherein the processor is further configured to use a measured value of afirst compressional wave in the formation, a second compressional wavein the formation, and a shear velocity of the formation to estimate thepermeability.